Decomposition Algorithms and Parallel Computing for Chance-constrained and Stochastic Integer Programs with Applications.

Abstract

The focus of this dissertation is to develop solution methods for stochastic programs with binary decisions and risk-averse features such as chance constraint or risk-minimizing objective. We approach these problems through scenario-based reformulations, which are often of intractable scale due to the use of a large number of scenarios to represent the uncertainty. Our goal is to develop specialized decomposition algorithms for solving the problem in reasonable time.

We first study a surgery planning problem with uncertainty in surgery durations. A common practice is to first assign operating rooms to surgeries and then to develop schedules. We propose a chance-constrained model that integrates these two steps. A branch-and-cut algorithm is developed, which exploits valid inequalities derived from a bin packing problem and single-machine scheduling problems. We also discuss models and solutions given ambiguous distributional information. Computational results demonstrate the efficacy of the proposed algorithm and provide insights into enhancing performance by the proposed model.

Next, we study general chance-constrained 0-1 programs, where decisions made before realization of uncertainty are binary. We develop dual decomposition algorithms that find solutions through bounds and cuts efficiently. We derive a proposition about computing the Lagrangian dual whose application substantially reduces the number of subproblems to solve, and deploy cut aggregation that accelerates the solution of subproblems. We also explore parallel schemes to implement our algorithms in a distributed system. All of them improve the efficacy effectively.

We then study dual decomposition for risk-averse stochastic 0-1 programs, which minimize the risk of some random outcome measured by a coherent risk function. Using generic dual representations for coherent risk measures, we derive equivalent risk-neutral minimax reformulations, to which dual decomposition methods apply. We investigate how to exploit the Lagrangian relaxation as the lower bounds by comparing three approaches. We also study parallelism more comprehensively, testing schemes that represent different combinations of basic/master-worker, synchronous/asynchronous and push/pull systems, and identify that the best is a master-worker, asynchronous and pull scheme, which achieves near-linear or even super-linear speedup.